WEBVTT
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so being given the point, ABC, with an
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equation of Z equals y squared minus X squared.
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We can plug in the point, um, and
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we get that C is equal to B squared minus
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a squared. So now we have the Z equals
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y squared minus X squared. That's how we end
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up getting the C equals B squared minus a squared
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. So now we want to plug in the Parametric
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equation into the equation of the hyperbolic parable oId and
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solve for C. So we have X equals a
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plus t. We have that y equals B plus
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T and Z equals C plus two times B minus
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80 then plugging in. Now that we're solving for
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C, we end up getting that C plus to
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be a T is equal to Z, but we
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can now write Z as B plus T squared minus
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a plus T squared. Then C is going to
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C plus two B T minus 2 80 is going
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to equal B squared plus two B t plus T
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squared minus a squared minus to a T minus t
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squared. And then we can combine like terms,
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and this is going to end up giving us a
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nice C equals B squared minus a squared. Well
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, that looks familiar because we found it up here
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as well, then plugging in a second Parametric equation
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. Um, we have X equals a plus t
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y equals B minus T and see again equals negative
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or sorry Z equals C minus two times be this
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time plus a team. We multiply everything out.
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We set Z equal to those other values. So
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now we have is a C minus to be T
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minus 2 80 equals B squared minus to B T
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plus T squared minus a squared minus 2 80 minus
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t squared. We subtract those things over, and
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once again, we end up getting C equals B
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squared minus a squared. So since both Parametric equations
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are equal to each other, then we can say
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that both of them lie entirely on Z equal to
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y squared minus X squared.